Comment on "Quasinormal modes in Schwarzschild-de Sitter spacetime: A simple derivation of the level spacing of the frequencies"

TL;DR

This paper critically examines the validity of extracting quasinormal modes using the first Born approximation, showing that the method's assumptions lead to invalid results for the imaginary parts of the frequencies.

Contribution

It demonstrates that the first Born approximation is mathematically unjustified for deriving quasinormal mode frequencies, challenging previous simple derivations.

Findings

First Born approximation constraints invalidate the pole-based QNM frequencies.

Inequalities restrict the imaginary parts of QNM frequencies, e.g., $0 \,\leq\, \omega_I < \kappa$.

Poles derived from the first Born approximation are not physically valid.

Abstract

It is shown here that the extraction of quasinormal modes (QNMs) within the first Born approximation of the scattering amplitude is mathematically not well founded. Indeed, the constraints on the existence of the scattering amplitude integral lead to inequalities for the imaginary parts of the QNM frequencies. For instance, in the Schwarzschild case, $0 \leq \omega_I < \kappa$ (where $\kappa$ is the surface gravity at the horizon) invalidates the poles deduced from the first Born approximation method, namely, $\omega_n = i n \kappa$.